Kernel Smoothing Regression

A non-parametrical regression (smoothing) tool using Gaussian kernel.

Description

Non-parametric regression is widely used in many scientific and engineering areas, such as image processing and pattern recognition.

Non-parametric regression is about to estimate the conditional expectation of a random variable:

E(Y|X) = f(X)

where f is a non-parametric function.

Based on the kernel density estimation technique, this code implements the so called Nadaraya-Watson kernel regression algorithm particularly using the Gaussian kernel. The default bandwidth of the regression is derived from the optimal bendwidth of the Gaussian kernel density estimation suggested in the literature. The code can also take care of missing data.

Acknowledgements

The author wishes to acknowledge the following in the creation of this submission:
Update PDF Estimation
This submission has inspired the following:
Multivariant Kernel Regression and Smoothing, Local Linear Kernel Regression, Volatility Surface

• MATLAB release: MATLAB 7.5 (R2007b)

[Dimitri Shvorob]

Nice work! There are some typos that can be corrected; with more time on their hands, one could add clarification/input check 'x must be a vector', allow an arbitrary set of evaluation points x, make chosen default bandwidth selection procedure more prominent (I have to say I didn't know this one; have you consulted Pagan's little book?), or allow an arbitrary kernel via a function-handle input argument. Do you want to try coding local polynomial (vs. local linear) kernel regression? :)

[Dimitri Shvorob]

Oops, I meant 'local polynomial (e.g. local linear) vs. local constant'. Does that make sense? Someone needs to refresh his stats :)

[Yi Cao]

Dimitri,

Thanks for your suggestions. The file has been updated to take some of your points: typos have been corrected (I hope I found all of them. BTW, I wish the MATLAB editor has a spelling check functionality. -:) ) and valid inputs have been properly checked.

In terms of better bandwidth, I do not aware any simple one available for such problem. Most I knew either too complcated or requires significant computation, such as the one through cross validation. If anyone know this, please leave a message here.

Option for user provided function handle is not implemented. Mainly, I am concerned the difficulty to check the correctness of the function specified (i.e. positiveness and integrating to 1).

Finally, local polynomial regression will require more work. It will be considered in a future version.

[Abel Brown]

this function rocks!! worked very well 'right outta the box'. My only question: Is is possible to do a variable bandwidth smoothing using this function?

At any rate, nice work
Thanks!

[J. Melon]

Hi Yi, wouldn't this function produce all NaN if the median of either X or Y is zero? (since h would be zero)

Is there anyway to fix it? for example, if sigma is zero, then let sigma = max(X) - min(X)

[Yi Cao]

Thanks for the comment. However, the zero median either in X or in Y should not be consider because we deal with a regression problem here. The case mentioned just means that X or Y are constant, then the regrassion problem is not well-posed and the solution is meaningless anyway.

[Shahab Anbarjafari]

Hi, nice work :)

[Xu Wings]

 

[John D'Errico]

 

[Kenneth Eaton]

 

[Michael Jordan]

 

[V. Poor]

 

[cf ]

It works very well and friendly.